University Heidelberg | SIP RG | Talks

Jan JOHANNES Sep 16,2015 Talks

Invited talk (.pdf):: conference Statistische Woche in Hamburg

Presented by:: Jan JOHANNES

Title:: Functional linear instrumental regression: Minimax-optimal estimation and adaptation

Abstract:: The estimation of a slope function β is considered in functional linear instrumental regression, where in the presence of a functional instrument W the dependence of a scalar response Y on the variation of an endogenous explanatory random function X is modelled by Y =<β,X>+σU, σ>0, for some error term U. Taking into account that the functional regressor X and the error term U are correlated in many economical applications, the random function W and the error term U are assumed to be uncorrelated. Given an iid. n-sample of (Y,X,W) a lower bound of the maximal mean integrated squared error is derived for any estimator of β over certain ellipsoids of slope functions. This bound is essentially determined by the mapping properties of the cross-covariance operator associated to the functional regressor X and the best linear predictor (BLP) of X given the instrument W. Assuming first that the BLP is known in advance a least squares estimator of β is introduced based on a dimension reduction technique and additional thresholding. It is shown that this estimator can attain the lower bound up to a constant under mild additional moment conditions. The BLP of X given the instrument W is generally, however, not known. Therefore, in a second step it is replaced by an estimator and sufficient conditions are provided to ensure the minimax-rate optimality of the resulting two stage least squares estimator. The results are illustrated by considering Sobolev ellipsoids and finitely or infinitely smoothing cross-covariance operators.

References:: H. Cardot and J. Johannes. Thresholding projection estimators in functional linear models. Journal of Multivariate Analysis, 101(2):395–408, 2010.; F. Comte and J. Johannes. Adaptive functional linear regression. The Annals of Statistics, 40 (6):2765–2797, 2012.; P. Hall and J. L. Horowitz. Nonparametric methods for inference in the presence of instru mental variables. The Annals of Statistics, 33(6):2904–2929, 2005.; J. Johannes. Functional linear instrumental regression. Technical report, Université catholique de Louvain, 2015.